Solve for $p$, $ \dfrac{6}{5p + 15} = \dfrac{9}{p + 3} + \dfrac{p - 4}{3p + 9} $
Solution: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5p + 15$ $p + 3$ and $3p + 9$ The common denominator is $15p + 45$ To get $15p + 45$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{6}{5p + 15} \times \dfrac{3}{3} = \dfrac{18}{15p + 45} $ To get $15p + 45$ in the denominator of the second term, multiply it by $\frac{15}{15}$ $ \dfrac{9}{p + 3} \times \dfrac{15}{15} = \dfrac{135}{15p + 45} $ To get $15p + 45$ in the denominator of the third term, multiply it by $\frac{5}{5}$ $ \dfrac{p - 4}{3p + 9} \times \dfrac{5}{5} = \dfrac{5p - 20}{15p + 45} $ This give us: $ \dfrac{18}{15p + 45} = \dfrac{135}{15p + 45} + \dfrac{5p - 20}{15p + 45} $ If we multiply both sides of the equation by $15p + 45$ , we get: $ 18 = 135 + 5p - 20$ $ 18 = 5p + 115$ $ -97 = 5p $ $ p = -\dfrac{97}{5}$